Applet below shows the magnitude condition |G(s)| = 1 plotted. For any given K, the condition is satisfied along a loop as shown by the red-blue boundary. To find the closed loop poles, one must find the point on this loop where the angle condition is satisfied.

For parameters (1.5, 1.0, 0.5, 2.0, sr, si) the magnitude condition is shown. The closed loop pole can be found by adjusting sr and si until a point on the loop |G(s)| = 1 is found where the phase is -180° and the gain = K. Adjusting sr, si shows that at sr = -0.34 and si = 1.04, K = ~2.02, f = ~-180.7°. Also at sr = -2.33 and si = 0.0, K = ~2.02 and f = -540°. The Closed loop poles are therefore at ~ -2.33 and ~-0.34 ± j1.04.

For parameters (1.0, 1.0, 1.0, 8.0, sr, si), K = 7.98 and f = -179.9° for sr = 0 and si = 1.73. Also K = 8.0, f = -540° with sr = -3.0 and si = 0.0. The closed loop poles are at -3.0 and ~±j1.73(correct value is 3

For parameters (3.5, 2.0, 0.2, 2.0, sr, si), K = 2.0 and f = -180.0° for sr = -0.9 and si = 0.0. K = 2.0, f = -180° with sr = -1.0 and si = 0.0. Also K = 2.02, f = -540° with sr = -3.81 and si = 0.0. The closed loop poles are therefore at ~-3.81, -1.0 and -0.9. ). In this case maximum K between the 2 poles to the right occur at sr = -0.95 which is the breakaway point from the real axis.

Gif image below shows how applet should appear.

COPYRIGHT © 2006 Cuthbert Nyack.